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Questions tagged [calculus-of-variations]

Optimization of functionals mostly defined on infinite-dimensional spaces.

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How to get the first variation of a complex lagrangian?

How do I get the first variation for this: $$ \int_C L(z,\phi,\frac{d\phi}{dz})dz$$ where: $$z=x+iy$$ $$dz=dx+idy$$ $$\phi=f(x,y)+ig(x,y)$$ and the integral is a complex line integral and $\phi$ is an ...
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Periodic motion on a torus and Lyusternik-Fet Theorem

I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold. For simplicity of description, take the 2-torus, and imagine it represents the configuration space ...
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1answer
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Let $X=\{u\in C^1[0,1]|u(0)=0\}$ and let $I:X\to\mathbb{R}$ be defined as $I(u)=\int_0^1 (u'^2-u^2)$. Which of the following is correct?

Let $X=\{u\in C^1[0,1]|u(0)=0\}$ and let $I:X\to\mathbb{R}$ be defined as $I(u)=\int_0^1 (u'^2-u^2)$. Which of the following is correct? $(a)$ I is bounded below $(b)$ I is not bounded below $(c)$ ...
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Is there a name for the minimal surface connecting two straight line segments in 3-dim Euclidean space?

In the $2$-dim Euclidean plane, the minimal curve that joins (and necessarily contains) two given points is a straight line. Here what is being minimized by the curve is the $1$-dim measure of the $1$...
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Relationship between Fundamental Lemma of Calculus of Variations and the Weighted-Residual Statement

The Weighted-Residual Method states that the integral of the Residual R(x) times the weighting function w(x) is equal to zero which means that R(x) = 0 On the other hand, Fundamental Lemma of ...
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Maximizing perimeter to area ratio of a function

I was trying to find the curve, $y = f(x)$, from $x = x_i$ to $x = x_f$, constrained by $y(x_i) = 0$ and $y(x_f) = 0$, such that the ratio between the arc length of the curve and the area below the ...
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1answer
21 views

Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$

Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$ Based on Euler-Lagrange(EL) equation, we have \begin{align} \frac{\partial xw'(x)}{\partial w(x)}-\...
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1answer
87 views

Which shape does an elastic rod take as its ends are getting closer?

Which shape does an idealized elastic rod take whose ends are moved towards each other? (The rod is supposed to be straight in relaxed state and to be deformable perpendicular to its direction, i.e. ...
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32 views

Computing the variation of $I[u] = \frac{1}{2}[u(1)]^2+\frac{1}{2}\int_0^1(u')^2\,dx$

I'm reading through Cassel's text on variational methods, and I'm struggling to follow Example 2.6 in the text (pg. 51). Example 2.6 proposes the problem of finding the stationary function of $$I[u] ...
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1answer
24 views

Why is the variation of a derivative the same as the derivative of the variation?

An example of this is with regard to the variation of the Lagrangian density $\mathcal{L}(\phi(x^{\mu}),\partial_{\mu}\phi)$: $$ \delta\mathcal{L}=\frac{\partial{\mathcal L}}{\partial\phi}\delta\phi+\...
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23 views

Minimizing Jensen-Shannon Divergence with constraints

I am trying to minimize the following function : $$J(p) = JSD(p_u || p)$$ with constraints : $$ \int p = 1 $$ $$ p(x) \geq \hat{\pi} p_p(x) $$ where $JSD$ is the Jensen Shannon Divergence, $p_u = \pi ...
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Is there any current method of distinguishing a local and a global extrema?

Given an arbitrary multivariate function $f(x)$ in $\mathbb{R}^n$ I was wondering if there is any method of distinguishing between a local minimum and a global minimum given you already have a point ...
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15 views

Jacobi identity for a given bracket operation

Let $F$, $G$ and $K$ be functionals of the form $$ F=\int\mathcal{F}(\phi_\alpha,\partial_i\phi_\alpha,\partial_i\partial_j\phi_\alpha,\dots,x^i,t)\,\text{d}v, $$ $$ G=\int\mathcal{G}(\phi_\alpha,\...
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What's up with the cycloid-shaped pot in Melville's Moby Dick?

People interested in the intersection between mathematics and fine literature may be familiar with the following quote from Herman Melville's famous novel Moby Dick: It is a place also for profound ...
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Noether on symmetry of a Weierstrass representation

Can someone explain which Weierstrass representation Emmy Noether refers to below? As context, Noether's second Conservation Theorem says if a Lagrangian has an infinite-dimensional Lie group of ...
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1answer
37 views

Integration by parts in the time derivative of a functional

I have a question regarding some calculations in Torres del Castillo's paper "Hamiltonian structures for classical fields". Let $\phi_{(a)}$ ($a=1,2,\dots,n$) be the variables that determine the state ...
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1answer
30 views

Positive Hausdorff measure and $L^{2}$ convergence. [closed]

i would like to know the relation between the positive Hausdorff measure and $L^{2}$ convergence in the ((1.3) existente theorem)
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35 views

Imposing a given constrain (f(x,y)>0) in a variational problem

My problem I am trying to solve a chemistry problem stating it as a constrained variational problem. I am learning this subject by myself and I have problems imposing a non-integral constrain. One ...
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Variation of Lagrangian with infinitesimal transformation of time and position variables

In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external ...
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Weak relative minimum

I've got a question about the following exercise: I'm supposed to prove that the functional $F(u):=\int_{-1}^1 x^2u'(x)^2 + xu'(x)^3 dx$ satisfies $\delta F(0,\zeta)=0 \; \forall\zeta\in C_0^1([-1,1])...
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calculus of variations: To find the angle at which the boundary point slides

Consider the functional $ I(y(x))=\int_{x_0}^{x_1}{f(x,y) \sqrt{1+y'^2} e^{\tan^{-1}{y'}}}dx$ where $f(x,y)\ne 0$. Let the left end of the extremal be fixed at the point $ A(x_0, y_0) $ and the ...
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1answer
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Minimizing Dirichlet energy defined on a open curve

As extension of this question, I was wondering what would be the Euler-Lagrange equations associated with the functional $$ E(u) = \frac{1}{2}\int_{\gamma} \lVert \nabla u \rVert^2 ds $$ The ...
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Calculus of variations ? electrostatic energy problem.

What is the maximum self-energy of an electrostatic distribution subject to the constraints that: the total charge is $1$; and the areal charge density anywhere is either $1$ or $0$. How ...
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1answer
27 views

Boundary conditions in minimizing Dirichlet energy for an image processing problem.

Suppose $$\mathcal{L} =\mathcal{L}(x,y,u,u_x,u_y) = \frac{1}{2} \lVert \nabla u \rVert^2$$ and I want to find $u$ such that the functional $$ E(u)=\int_{\Omega} \mathcal{L}dxdy $$ is minimized, ...
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1answer
38 views

Sobolev Inequality of Uniform Norm - Proof Explanation

In Brezis book, in the chapter of the Sobolev spaces, there is a proof for the inequality \begin{gather} \| u \|_{L^{\infty}(I)} \leq C \| u \|_{W^{1.p}(I)} \end{gather} for every $u \in W^{1,p}(I)$ ...
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2answers
54 views

Numerical compuation of functional derivative

I was wondering if it is possible to numerically compute $$\frac{\delta L}{\delta f}$$ where $L$ is a functional of the function $f$ known only in a discrete number of point by numerical computation ...
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Tautochrone and Brachistochrone problems are equivalent

It is a well known fact that the brachistochrone (the problem of finding the curve of quickest descent in a uniform gravitational field) and the tautochrone (the problem of finding a curve from which ...
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Variation and functional derivative

Let $F$ be a functional on a certain space of functions (actually I don't know what's necessary to define functional derivative). Is the following equality true for any small variation $\delta f(x)$ ? ...
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1answer
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Calculus of Variations: Hamilton’s canonical equations

In Calculus of Variations, Hamilton’s canonical equations (Calculus of Variations and Optimal Control Theory by Daniel Liberzon, p. 45) are $$y'~=~H_p,\tag{1}$$ $$p'~=~-H_y.\tag{2}$$ I understand ...
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Constrained equilibrium for inflated sheet

I am trying to figure out the shape that a disc of uniformly elastic material makes if you fix its circumference firmly to a flat table and slowly fill the interior with air. So far, I've determined ...
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Minimum representation of total generalized variation

I need help understanding the proof of Theorem 3.1 from this paper. To make this post self-containded, at first I briefly review the definitions. Let $\Omega \subset\mathbb{R}^d$ be a bounded ...
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Can path integral approach be applied to continous intertemporal optimization (e.g. in economics)?

Economics have this intertemporal optimization for the life-time consumption/work choice utility maximization http://www.waelde.com/pdf/AIO.pdf . Can path integral (functional integral, variational ...
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Limit Definition of Functional Derivative for multifunction functionals

Imagine variational calculus on nonlinear multifunction functionals. Let $C^k[a,b]$ denote the set of all $k$ times differentiable functions from $[a,b]$ to $\mathbb{R}$. Now consider a functional $F[...
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1answer
54 views

Limit of a sequence of functions in the Sobolev space

Let $\mathbb{D} \subset \mathbb{R}^2$ be the unit disk and consider the Dirichlet problem of the Laplace equation $\Delta u = 0$ in $\mathbb{D}$ with $u = 0$ on $\partial \mathbb{D}$. Then we know ...
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1answer
33 views

The connection between Image Segmentation with Neural Networks versus graph cut and mumford-shah type approaches

This question is meant to be a more technical question, so if I have phrased it as soliciting an opinion, any suggested language updates would be appreciated. I am relatively new to image processing,...
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37 views

Calculus of Variations. Finding the extremals of a perturbed Lagrangian

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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Minimising the Forward KL-divergence via moment matching

The forward KL-divergence is defined as \begin{align}\label{eq:for_KL2} \text{KL}(p(\theta|x)||q(\theta)) & = \int p(\theta|x) \ln \left\{ \frac{p(\theta|x)}{q(\theta)}\right\} dZ\\ & = \...
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Find a Function that Maximizes Another Function With an Integral

I would like to find a function $z(y)$ which maximizes the expression $$\big(y - p\,z(y)\big)\left(\alpha z(y) - \int_a^b z(y)f(y)\,dy\right),$$ but I am not too sure how to go about this. My ...
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Variational Problem, Role of boundary conditions in derivation of Euler-Lagrange?

This is something of a sanity check as I don't have that much formal background in the calculus of variations. My intuition to the block-quoted question below is that the answer is affirmative but I'm ...
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Understanding domains and codomains in derivation of Euler-Lagrange equations

I am trying my best to understand the derivation of Euler-Lagrange equations but I am completely lost even at understanding what kind of inputs and outputs do we have. For you to understand my ...
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Constructing a differential equation involving local isometries

I bet there must be some research about this topic I'm not able to find. Suppose I have a smooth surface patch $\sigma_0$ I want to construct a local isometry $\sigma_{\infty}$ with some specific ...
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What does it really mean for a Lagrangian to be independent of the base coordinates?

In this question I will be considering what is called in physics a "classical Lagrangian field theory" from a geometric point of view. One is given an $n$ dimensional (smooth, real) "spacetime ...
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1answer
36 views

“Cyclic” Variables

This post may belong in the physics StackExchange site, but I'd to hear an answer from a mathematician rather than a physicist. I did find a few answers for this question there, but none made a whole ...
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35 views

Find stationary values of a functional defined by integrating part of several variables

Is there a way to find stationary values of a functional defined by integrating part of several variables (the others are given), like this $$ J[u;x_{m+1},\cdots,x_{n}] = \int \cdots \int F(x_1,\cdots,...
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1answer
25 views

Deriving Classical Equations of Motion for Chern-Simons Theory

I am reading the Wikipedia article on Chern-Simons theory https://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory, where it states that the action of Chern-Simons theory is $S=\frac{k}{4\pi}\int_M ...
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Under which conditions we can use the evolution equation to find the solution of a variational problem?

I'm looking for a reference that can explain under which conditions the variational problem $$ E = \int_{\Omega} \mathcal{L}d\Omega $$ can be solved using the evolution equation $$ \mathcal{\...
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Relationship between energy functions in calculus of variations.

I've been implementing lately few algorithms based on energy functions of the form. $$ E = \int_\Omega \mathcal{L}(x,f,f')dx $$ In the above the cost function $E$ defines an actual energy that we ...
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Use a varied function to show that a straight line is the minimum distance between 2 points.

I have the following problem, and I don't know where to begin. Consider the curve connecting $(x_0,y_0) = (0,0)$ and $(x_1,y_1) = (1,1)$. Show by explicit computation that the function $y(x) =x$ ...
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2answers
62 views

Linear approximation of membrane energy

From a reference I have in mesh processing the following well known equation is used to compute the area of a parametric surface $$ A(\sigma) = E_M(\sigma) = \int_U \sqrt{EF - G^2}dU $$ Which is ...
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Is there a trick to do calculus of variations with bounded functions?

Suppose I have a calculus of variation problem. That is to say a functional of the form $f(t) \mapsto \int_0^1 L \big (t,f(t),f'(t) \big ) dt$ which I want to find extrema over all differentiable ...